The rationality problem for fields of invariants under linear algebraic groups (with special regards to the Brauer group)

TL;DR

This survey explores the longstanding open problem of whether the field of invariants under a reductive algebraic group action is purely transcendental, discussing rationality notions, quotient constructions, and the unramified Brauer group with recent computational insights.

Contribution

It provides a comprehensive overview of the rationality problem for invariant fields, including recent work on the unramified Brauer group by Saltman and Bogomolov, and discusses specific cases and open questions.

Findings

Analysis of rationality and quotient notions

Descriptions of unramified Brauer group computations

Discussion of open problems for connected reductive groups

Abstract

This is a survey on the ancient question : Let G be a reductive group over an algebraically closed field k and let V be a vector space over k with an almost free linear action of G on V. Let k(V) denote the field of rational functions on V. Is the subfield of G-invariants of k(V) purely transcendental over k ? For G connected, this is still an open question. After a discussion of general matters (various notions of rationality, various notions of quotients, the no-name lemma), we consider several specific groups G. We then discuss the unramified Brauer group of a function field and describe the work of Saltman and of Bogomolov, leading to computations of the unramified Brauer group of fields of G-invariants. The text is a thoroughly revised version of a text distributed at the 9th latino-american school (Santiago de Chile, July 1988), various versions of which had been circulated over the years. ----- Soient k un corps alg'ebriquement clos, V un espace vectoriel sur k et G un groupe r'eductif connexe sur k agissant lin'eairement sur V. Supposons l'action g'en'eriquement libre. Le sous-corps des G-invariants du corps des fonctions rationnelles k(V) est-il transcendant pur ? Le pr'esent texte est un rapport g'en'eral sur cette vieille question, encore ouverte lorsque G est connexe. On d'ecrit en particulier des travaux de Saltman et de Bogomolov sur le groupe de Brauer non ramifi'e des corps de G-invariants. Ce texte est une version tr`es remani'ee d'un texte distribu'e `a la neuvi`eme 'ecole d''et'e latino-am'ericaine (Santiago de Chile, Juillet 1988).